The spectra of graphs based on general graph operations

Abstract

For two graphs $G_1$ and $G_2$. Let $\mathcal{B}$ be a set of non-zero binary 2-tuples, i.e. $\mathcal{B}\subseteq \{0,1{\}}^2\setminus \{(0,0)\}$. The NEPS (non-complete extended p-sum) of graphs $G_1,G_2$ with basis $\mathcal{B}$ is denoted by NEPS$(G_1,G_2;\mathcal{B})$. Let $G_1[G_2]$ be the lexicographic product of $G_1$ and $G_2$. In this paper, we determine the spectra of the graph $G_0{◼}_{k_1}$NEPS$(G_1,G_2;\mathcal{B})$ and $G_0{◼}_{k_1}{G}_1[G_2]$ with regular graphs $G_0$, $G_2$ and an arbitrary graph $G_1$ in terms of their spectra. As applications, the results on spectra enable us to construct some new cospectral graphs and integral spectrum graphs.

Keywords:

• Spectrum
• NEPS of graphs
• lexicographic product
• integral spectrum graphs
• cospectral graphs

AMS classification:

• 05C50

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their valuable comments and helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the Natural Science Foundation of Xinjiang Province [grant number 2021D01C069], the National Natural Science Foundation of China [grant number 12161085] and the Scientific Research Plan of Universities in Xinjiang, China [grant number XJEDU2021I001].